Properties

 Label 245.2.b.d Level $245$ Weight $2$ Character orbit 245.b Analytic conductor $1.956$ Analytic rank $0$ Dimension $2$ CM discriminant -35 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,2,Mod(99,245)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("245.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 q^{4} + \beta q^{5} - 2 q^{9}+O(q^{10})$$ q + b * q^3 + 2 * q^4 + b * q^5 - 2 * q^9 $$q + \beta q^{3} + 2 q^{4} + \beta q^{5} - 2 q^{9} - 3 q^{11} + 2 \beta q^{12} - 3 \beta q^{13} - 5 q^{15} + 4 q^{16} + \beta q^{17} + 2 \beta q^{20} - 5 q^{25} + \beta q^{27} + 9 q^{29} - 3 \beta q^{33} - 4 q^{36} + 15 q^{39} - 6 q^{44} - 2 \beta q^{45} - 5 \beta q^{47} + 4 \beta q^{48} - 5 q^{51} - 6 \beta q^{52} - 3 \beta q^{55} - 10 q^{60} + 8 q^{64} + 15 q^{65} + 2 \beta q^{68} - 12 q^{71} - 6 \beta q^{73} - 5 \beta q^{75} + q^{79} + 4 \beta q^{80} - 11 q^{81} + 4 \beta q^{83} - 5 q^{85} + 9 \beta q^{87} - 3 \beta q^{97} + 6 q^{99} +O(q^{100})$$ q + b * q^3 + 2 * q^4 + b * q^5 - 2 * q^9 - 3 * q^11 + 2*b * q^12 - 3*b * q^13 - 5 * q^15 + 4 * q^16 + b * q^17 + 2*b * q^20 - 5 * q^25 + b * q^27 + 9 * q^29 - 3*b * q^33 - 4 * q^36 + 15 * q^39 - 6 * q^44 - 2*b * q^45 - 5*b * q^47 + 4*b * q^48 - 5 * q^51 - 6*b * q^52 - 3*b * q^55 - 10 * q^60 + 8 * q^64 + 15 * q^65 + 2*b * q^68 - 12 * q^71 - 6*b * q^73 - 5*b * q^75 + q^79 + 4*b * q^80 - 11 * q^81 + 4*b * q^83 - 5 * q^85 + 9*b * q^87 - 3*b * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 - 4 * q^9 $$2 q + 4 q^{4} - 4 q^{9} - 6 q^{11} - 10 q^{15} + 8 q^{16} - 10 q^{25} + 18 q^{29} - 8 q^{36} + 30 q^{39} - 12 q^{44} - 10 q^{51} - 20 q^{60} + 16 q^{64} + 30 q^{65} - 24 q^{71} + 2 q^{79} - 22 q^{81} - 10 q^{85} + 12 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 - 4 * q^9 - 6 * q^11 - 10 * q^15 + 8 * q^16 - 10 * q^25 + 18 * q^29 - 8 * q^36 + 30 * q^39 - 12 * q^44 - 10 * q^51 - 20 * q^60 + 16 * q^64 + 30 * q^65 - 24 * q^71 + 2 * q^79 - 22 * q^81 - 10 * q^85 + 12 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 2.23607i 2.23607i
0 2.23607i 2.00000 2.23607i 0 0 0 −2.00000 0
99.2 0 2.23607i 2.00000 2.23607i 0 0 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
5.b even 2 1 inner
7.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.b.d 2
3.b odd 2 1 2205.2.d.h 2
5.b even 2 1 inner 245.2.b.d 2
5.c odd 4 2 1225.2.a.q 2
7.b odd 2 1 inner 245.2.b.d 2
7.c even 3 2 245.2.j.b 4
7.d odd 6 2 245.2.j.b 4
15.d odd 2 1 2205.2.d.h 2
21.c even 2 1 2205.2.d.h 2
35.c odd 2 1 CM 245.2.b.d 2
35.f even 4 2 1225.2.a.q 2
35.i odd 6 2 245.2.j.b 4
35.j even 6 2 245.2.j.b 4
105.g even 2 1 2205.2.d.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.b.d 2 1.a even 1 1 trivial
245.2.b.d 2 5.b even 2 1 inner
245.2.b.d 2 7.b odd 2 1 inner
245.2.b.d 2 35.c odd 2 1 CM
245.2.j.b 4 7.c even 3 2
245.2.j.b 4 7.d odd 6 2
245.2.j.b 4 35.i odd 6 2
245.2.j.b 4 35.j even 6 2
1225.2.a.q 2 5.c odd 4 2
1225.2.a.q 2 35.f even 4 2
2205.2.d.h 2 3.b odd 2 1
2205.2.d.h 2 15.d odd 2 1
2205.2.d.h 2 21.c even 2 1
2205.2.d.h 2 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}$$ T2 $$T_{3}^{2} + 5$$ T3^2 + 5 $$T_{19}$$ T19

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 5$$
$5$ $$T^{2} + 5$$
$7$ $$T^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 45$$
$17$ $$T^{2} + 5$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 125$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 180$$
$79$ $$(T - 1)^{2}$$
$83$ $$T^{2} + 80$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 45$$